Arithmetic geometry · algebraic geometry · algebraic cycles · motives

Tate Conjecture

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Over a finite field, do algebraic cycles account for every Frobenius-fixed even cohomology class, with no hidden nonsemisimple behavior at the corresponding eigenvalue?

{cl}_{ℓ, n} : {CH}^{r} (X_{F_{q^{n}}}) \otimes Q_{ℓ} ↠ H^{2 r} (\bar{X}, Q_{ℓ} (r))^{F^{n} = 1}, F^{n} semisimple at 1

Known results and sources

The strong finite-field Tate conjecture asks whether every Frobenius-fixed even cohomology class comes from an algebraic cycle and whether the relevant Frobenius block is semisimple.

Research problem

Exact mathematical statement

Let $X/\mathbf F_q$ be smooth and projective, let $r$ be a codimension, let $\ell\nmid q$ be prime, and let $k_n=\mathbf F_{q^n}$ be any finite extension. The strong finite-field Tate conjecture asserts that the cycle map

\operatorname{cl}_{\ell,n}:CH^r(X_{k_n})\otimes\mathbf Q_\ell\longrightarrow H^{2r}(\bar X,\mathbf Q_\ell(r))^{F^n=1}

is surjective and that $F^n$ is semisimple at eigenvalue $1$ on the twisted group, equivalently at eigenvalue $q^{nr}$ on untwisted degree- $2r$ cohomology. the source targets this all-varieties, all-codimensions finite-field form and explicitly does not claim a complete proof.

Problem infographic

Problem at a glance

A problem-first scientific diagram starts with a smooth projective variety X over a finite field, sends codimension-r algebraic cycles through the Tate cycle map into twisted degree-2r cohomology, isolates the vectors fixed by the nth Frobenius power, and marks both surjectivity and semisimplicity at eigenvalue one as open in general. — For every smooth projective variety over a finite field, every codimension, every admissible prime, and every finite extension, the conjecture predicts cycle-map surjectivity and semisimplicity at the Tate eigenvalue; the full statement remains open.

Current mathematical picture

Where work on Tate Conjecture stands

Partially resolved

Selected route highlights from the current work. This is not yet a complete mathematical inventory.

Useful failureSource-reported limitation

The inference is false: in $\operatorname{Rep}_{\mathbf Q_\ell}(\mu_2)$ , the nontrivial sign line $Q$ has $Q^{\otimes2}\simeq\mathbf1$ , $F_Q=1$ , and $\operatorname{Hom}(\mathbf1,Q)=0$ . Every even tensor power passes while $Q$ is not the unit, so even-tensor propagation can address Gate I but cannot eliminate Gate II. The route closes only if every normalized block has the full m-squared-dimensional numerical endomorphism algebra and every geometric even line Q with Q tensor-square equal to the unit, trivial Frobenius, and an effective Lefschetz twist is itself the unit. The current work requires an independent checking of the reduction before those gates are treated as publication-ready.

Route status · Narrowed route

Main reductionFrobenius scalarizes numerically

On the numerical image of the generalized Tate-primary block, the current work reports exact scalar Frobenius action by q to the r without assuming cohomological semisimplicity.

Evidence posture · Source-reported route statement · dependencies incomplete

Priority open bridgeIndependently audit the parity–trace lifting argument and the complete matrix-defect/quadratic-line theorem with all category, coefficient, duality, and Krull–Schmidt details.Task status · Work already reported in progress

Research-record correctionResearch-record correction

We corrected supporting details in the research record. The mathematical claims and their status did not change.

Reader-facing record corrected; mathematics unchanged

Work mapped so far

Tate Conjecture in numbers

1.1kretained lines of mathematical investigation1,075 in the current working snapshot

Argument development: 856 · 80%
Explored or eliminated routes: 14 · 1%
Computational analysis: 108 · 10%
Open obligations: 18 · 2%
Definitions and setup: 79 · 7%

9selected mapped statements1routes investigated3open questions2contribution-ready tasks

How this is measured

This measures retained mathematical investigation, not proximity to a proof. Code, data, logs, repeated text, operational instructions, and generated presentation copy are excluded.

Proposed byJohn TateProposed1963Status checkedAug 7, 2026

Source recordedAug 7, 2026

Recommended next task

Prove Gate I: every normalized middle block has m-squared independent numerical endomorphisms.

Suggested move: Formalize the trace-pairing certificate and search for geometric correspondences giving matrix units, while testing every purely tensor-formal argument against arbitrary semisimple representation categories.

What would count as progress

Retain an exact proof or counterexample for the stated subproblem.

Argument map and routes

How the current approaches connect

Claims, reductions, open questions, active routes, and narrowed alternatives in one mathematical map.

Visible working map

Research route map

13 selected steps

Selected claims, active routes, useful failures, and open questions from the current research map. Arrows appear only for explicitly recorded relationships.

Scroll horizontally to explore the route

Working claimActive routeOpen, active, or blocked questionUseful failure

Working overview, not proof. The map shows selected recorded relationships; more nodes or edges do not establish correctness or completion.

Explored alternatives

Other routes

1 recorded

Narrowed routeSource-reported limitation

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

3 featured tasks

Prove Gate I: every normalized middle block has m-squared independent numerical endomorphisms.Suggested move: Formalize the trace-pairing certificate and search for geometric correspondences giving matrix units, while testing every purely tensor-formal argument against arbitrary semisimple representation categories.

Ready to work on

Prove Gate II: the residual even quadratic line with trivial Frobenius and effective Lefschetz twist is trivial.Suggested move: Attack the multiplicity-one case first through effectivity cancellation or rank-two finite-etale rigidity, retaining finite-extension descent and the non-Artin possibility explicitly.

Ready to work on

Independently audit the parity–trace lifting argument and the complete matrix-defect/quadratic-line theorem with all category, coefficient, duality, and Krull–Schmidt details.Suggested move: Write a standalone proof for one normalized block, check every surjection and lifting step, and stop at the first unverified implication rather than importing the current work's summary as authority.

Work already reported in progress

Sourced mathematical context

The known mathematical landscape

Context collected Aug 7, 2026

Current statusPartially resolved

The rational Tate conjecture is proved for major classes, including divisors on abelian varieties over finite fields, the corresponding abelian-variety homomorphism theorem over number fields, and divisors on K3 surfaces in every positive characteristic. Arbitrary codimension for arbitrary smooth projective varieties over finite fields remains wide open.

[5][6][7]

External progress

What the literature has established

Selected external milestones in reverse chronological order, with their evidence posture.

2026
Peer reviewedBalkan and Schreieder reduced the joint Tate, Beilinson, and Grothendieck–Serre semisimplicity package over finite fields to vanishing of a natural birational invariant on projective space.[9]
2015–2016
Peer reviewedKuga–Satake and integral-model methods completed the divisor theorem for K3 surfaces in odd characteristic and characteristic 2.[7][8]
1985
Peer reviewedNygaard and Ogus proved the Tate conjecture for K3 surfaces of finite height over finite fields, extending the ordinary K3 case.[4]
1983
Peer reviewedFaltings proved semisimplicity and the l-adic homomorphism theorem for abelian varieties over number fields as part of his finiteness work.[3]

Mathematical neighborhood

Related results and reusable starting points

Current focusTate conjecture

Equivalent formulationStrong pole-order form of the Tate conjecture

Over finite fields, after including the needed semisimplicity and numerical-versus-homological conditions, the cohomological formulation matches a pole-order statement for the zeta function. The qualifications are essential.

[5][6]

Logical consequenceGrothendieck standard conjectures and semisimplicity

The Tate conjecture implies important algebraicity statements among Grothendieck's standard conjectures; together with semisimplicity it helps identify homological and numerical equivalence.

[5][9]

Formal and computational footholds

Existing statements, libraries, computations, and datasets that can shorten the next serious attempt.

formal library support · partial resource linkedmathlib algebraic cycles on schemes
Mathlib defines algebraic cycles on schemes and a pushforward API. It does not by itself provide the cycle-class map or the Tate surjectivity statement.
[10]
formal library support · partial resource linkedmathlib l-adic cohomology on the pro-étale site
Mathlib defines l-adic sheaves and cohomology groups on the pro-étale site. Its documentation notes that comparison with classical étale cohomology is future work, and no statement-aligned Tate conjecture was verified.
[11]
dataset · source linked; not reproduced by ProofAtlasLMFDB abelian varieties over finite fields
The LMFDB collection records Weil polynomials, Frobenius data, slopes, endomorphism-related fields, and point counts for bounded ranges of finite-field abelian-variety isogeny classes. It supports examples in a major solved class but is not evidence for the unrestricted conjecture.
[12]
software · source linked; not reproduced by ProofAtlasSageMath finite-field scheme zeta functions
SageMath exposes zeta-function and zeta-series computations for schemes and curves over finite fields. Frobenius multiplicities can suggest the expected cycle rank, but constructing matching algebraic cycles is the substantive conjectural step.
[13]

Formalization opportunities

Lean work can make these reusable foundations precise without being presented as a proof of the core problem.

Formalization targetA checked cycle-class map from codimension-r algebraic cycles with rational coefficients to l-adic cohomology in degree 2r.
Formalization targetA formal Galois action, Tate twist, fixed-subspace construction, and comparison with the chosen pro-étale or classical étale cohomology model.
Formalization targetFormal smoothness, projectivity, geometric base change, and finitely-generated-base-field hypotheses connected to the cohomology and cycle APIs.
Formalization targetA statement-level separation of the rational, integral, strong pole-order, and semisimplicity variants so that no stronger or false integral assertion is substituted.

Research-record corrections

What changed in the research record

These notes describe corrections to cited passages, highlighted tasks, or connections between claims. The mathematical claims and their status did not change.

Research-record correctionWe corrected supporting details in the research record. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details

The initial argument structure appears separately. Uploads, model runs, and presentation changes do not count as mathematical updates.

Detailed research inventory

Claims, milestones, and routes in the current map

This view highlights the mathematical statements most useful for following the current route.

6 standing statements3 proposed statements3 open questions1 narrowed routes

Statements by mathematical role9 selected mapped statements

theorem candidate1 of 91
reduction2 of 92
lemma3 of 93
equivalence2 of 92
negative result1 of 91

Selected mathematical clusters3 mathematical clusters

Statements and reductionsClaims, implications, and derivations in the current map.17 displayed rows

retained route statementEvery Frobenius-fixed Tate class should be algebraic, with semisimple Frobenius behavior on the full Tate block.
retained route statementCurrent reductionintermediate
retained route statementClosing targetintermediate
retained route statementCanonical Tate-primary blockintermediate
retained route statementFrobenius scalarizes numericallyintermediate
retained route statementParity–trace lifting lemmaintermediate
retained route statementNumerical Tate-atom reductionintermediate
retained route statementDefect zero detects the endomorphism motiveintermediate
retained route statementTwo exact open gatesintermediate
Recorded relationshipThe source material reports this as a route toward the conjecture; missing or unaudited premises remain and the reduction does not itself prove the target.supports · reported by source
Recorded relationshipthis work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipthis work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipthis work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipthis work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipthis work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipthis work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
DerivationThe current work reports that completing the closing target would advance the reduction to the main conjecture; this remains an informal route, not a verified derivation.proposed

Open questionsSpecific obligations that remain open in the current routes.3 displayed rows

Research targetIndependently audit the parity–trace lifting argument and the complete matrix-defect/quadratic-line theorem with all category, coefficient, duality, and Krull–Schmidt details.in progress reported
Research targetProve Gate I: every normalized middle block has m-squared independent numerical endomorphisms.open
Research targetProve Gate II: the residual even quadratic line with trivial Frobenius and effective Lefschetz twist is trivial.open

Explored routes and evidenceChallenges, computations, and approaches that have already narrowed the search.3 displayed rows · 1 route included

Useful failureSource-reported limitationreported failure
ComputationThe retained source supplies a publication-level audit checklist rather than a completed independent checking; this page records no implementation, input, or certificate digest.The listed reductions remain source-reported narrative mathematics pending the source's independent checking checklist. this page supplies no independent proof, formal verification, or acceptance of the Tate conjecture. · reported unreproduced
Narrowed routeSource-reported limitationThe inference is false: in $\operatorname{Rep}_{\mathbf Q_\ell}(\mu_2)$ , the nontrivial sign line $Q$ has $Q^{\otimes2}\simeq\mathbf1$ , $F_Q=1$ , and $\operatorname{Hom}(\mathbf1,Q)=0$ . Every even tensor power passes while $Q$ is not the unit, so even-tensor propagation can address Gate I but cannot eliminate Gate II. The route closes only if every normalized block has the full m-squared-dimensional numerical endomorphism algebra and every geometric even line Q with Q tensor-square equal to the unit, trivial Frobenius, and an effective Lefschetz twist is itself the unit. The current work requires an independent checking of the reduction before those gates are treated as publication-ready.

How to interpret these counts

A statement may be a lemma, conditional reduction, special case, documented limitation, or open target. These counts describe the work's structure; they do not estimate distance to a proof.

Research outlook

Conditions that would advance the current route

Priority open bridgeIndependently audit the parity–trace lifting argument and the complete matrix-defect/quadratic-line theorem with all category, coefficient, duality, and Krull–Schmidt details.

1 approach has already been tested and narrowed. The task above is the current priority within the larger open route.

Evidence needed nextConcrete conditions for progress

A result can change the outlook by closing the bridge, narrowing its scope, or showing that the route cannot work.

Supply a complete argument with every imported premise identified.
Survive an independent attempt to falsify the proposed step.

Continue the mathematics

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Prepared starting pointProve Gate I: every normalized middle block has m-squared independent numerical endomorphisms.

Tate Conjecture · ready to start

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Research contextPrepared context for any AI agent

Over a finite field, do algebraic cycles account for every Frobenius-fixed even cohomology class, with no hidden nonsemisimple behavior at the corresponding eigenvalue?

Exact question and boundaries
Current routes and known obstacles
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Sources and references14 cited works · next context review by Nov 7, 2026

The mathematical context was checked on Aug 7, 2026. Status can be refreshed sooner after a material result or claim.

1
Algebraic cycles and poles of zeta functionsoriginal source · John Tate · Arithmetical Algebraic Geometry, Harper & Row · 1965 · accessed Aug 7, 2026
2
Endomorphisms of Abelian Varieties over Finite Fieldspeer reviewed result · John Tate · Inventiones Mathematicae · 1966 · accessed Aug 7, 2026
3
Endlichkeitssätze für abelsche Varietäten über Zahlkörpernpeer reviewed result · Gerd Faltings · Inventiones Mathematicae · 1983 · DOI 10.1007/BF01388432 · accessed Aug 7, 2026
4
Tate's conjecture for K3 surfaces of finite heightpeer reviewed result · Niels Nygaard, Arthur Ogus · Annals of Mathematics · 1985 · accessed Aug 7, 2026
5
Conjectures on algebraic cycles in l-adic cohomologyoriginal source · John Tate · American Mathematical Society, Proceedings of Symposia in Pure Mathematics 55.1 · 1994 · DOI 10.1090/pspum/055.1/1265523 · accessed Aug 7, 2026
6
The Tate conjecture over finite fieldssurvey or monograph · James S. Milne · American Institute of Mathematics workshop notes · 2007; revised 2008 · ARXIV 0709.3040 · accessed Aug 7, 2026
7
The Tate conjecture for K3 surfaces in odd characteristicpeer reviewed result · Keerthi Madapusi Pera · Inventiones Mathematicae · 2015 · ARXIV 1301.6326 · DOI 10.1007/s00222-014-0557-5 · accessed Aug 7, 2026
8
2-adic integral canonical models and the Tate conjecture in characteristic 2peer reviewed result · Wansu Kim, Keerthi Madapusi Pera · Forum of Mathematics, Sigma · 2016-10-05 · ARXIV 1512.02540 · DOI 10.1017/fms.2016.23 · accessed Aug 7, 2026
9
Cycle conjectures and birational invariants over finite fieldspeer reviewed result · Samet Balkan, Stefan Schreieder · Selecta Mathematica · 2026-04-01 · DOI 10.1007/s00029-026-01142-0 · accessed Aug 7, 2026
10
Mathlib.AlgebraicGeometry.AlgebraicCycle.Basicformalization · mathlib contributors · Lean mathematical library · accessed Aug 7, 2026
11
Mathlib.AlgebraicGeometry.Sites.ElladicCohomologyformalization · mathlib contributors · Lean mathematical library · accessed Aug 7, 2026
12
Isogeny Classes of Abelian Varieties over Finite Fields in the LMFDBsoftware or dataset · Taylor Dupuy, Kiran Kedlaya, David Roe, Christelle Vincent · L-functions and Modular Forms Database · 2020 · ARXIV 2003.05380 · accessed Aug 7, 2026
13
Schemes: zeta_function and zeta_series over finite fieldssoftware or dataset · Sage Development Team · SageMath Reference Manual 10.8 · accessed Aug 7, 2026
14
Tate conjectureencyclopedia · Wikimedia Foundation · accessed Aug 7, 2026

Important qualifications

The status judgment concerns the rational l-adic Tate conjecture; integral variants and their counterexamples are outside that judgment.
The phrase Tate conjecture for abelian varieties is codimension-sensitive: Tate's and Faltings' homomorphism theorems establish the divisor or endomorphism case, not arbitrary higher-codimension cycles on every abelian variety.
The K3 milestones concern divisors and do not prove arbitrary-codimension statements on arbitrary varieties.
The 2026 birational-invariant theorem is an equivalence involving Tate, Beilinson, and semisimplicity conjectures, not a resolution of them.
No unreviewed source material or packet-provided proof or computation was read.
The scoped formalization search verified public mathlib prerequisites but found no complete statement-aligned Tate conjecture; this does not prove global absence.
LMFDB and Sage resources were inspected through public pages and documentation but were not rerun or independently reproduced.

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